I've never been the best with combinatorics, but I could always at least get by. I thought of a problem that has really sparked an interest with me, though.
(I will start with a specific example, then move on to generalize)
Say you are doing laundry and get to the part where you are matching socks. Assume that you have 4 pairs of white socks, 3 pairs of black socks, 2 pairs of red socks, and 1 pair of yellow socks. You proceed to match socks in the following way:
You start with all of the socks in a basket (and every sock has a matching pair!). You randomly pull the first sock out and lay it in front of you. One at a time, you randomly pull each sock from the basket and do one of two things. If there is no sock laying in front of you that matches that sock, you also lay it in front of you. However, if there is a sock laying in front of you that matches it, you put those two together and put them away into your drawer. (Here, we assume that any two white socks will match each other, any two black sock will match each other, etc.)
What is the expected value of the largest number of socks you will have in front of you throughout this entire process?
To generalize, say we have \(a_1\) pairs of socks of the first color, \(a_2\) pairs of socks of the second color, up to \(a_n\) pairs of socks of the \(n\)th color, and you match your socks by the same method described above. What is the expected value of the largest number of socks you will have laying in front of you for this process? Is there a general method to determine this expected value?
I have attempted this question myself, but find my mediocrity within the subject of Combinatorics to prove to strong for me to work with more than 3 different colors of socks. Any help?